Lemon meringue pie that weeps tastes gummy or soggy, rather than crispy and chewy. Meringue can cascade off the pie from excess water. Once you know why meringue weeps, you can control those different variables to achieve success with this pie every time you make it.
If you make meringue with old eggs, it may weep. Use fresh eggs to guard against this. Separate your eggs one at a time, and don't use an egg white if you accidentally get some yolk in there. Egg whites with yolk bits won't whip properly.
When beating the egg whites for meringue, either overbeating or underbeating can give you an unstable meringue. Meringue made from overwhipped egg whites can "pop" when it gets baked, causing weeping. Meringue made from underwhipped egg whites will also leak, because the mixture is not strong enough to hold up. For success with meringue, add a pinch of cream of tartar into the egg whites when they become frothy, during the first stage of beating. This acid helps strengthen egg white proteins. Also, drizzle the sugar in slowly a little at a time, which helps the meringue slowly build up bubbles.
Topping hot pie with meringue, then baking to set the meringue, gives you a better shot at a no-weep pie. Steam from a hot filling will pass through the meringue. By contrast, if you add meringue to a cooled pie, then reheat the pie and the meringue, that steam will exit the pie much slower, weighing down the meringue with moisture. This trapped moisture weeps, and can cause meringue to slide off your pie.
Meringue for pies isn't meant to last forever, so even the strongest meringue will break down. Refrigerating your lemon meringue pie before baking speeds up weeping. For a no-weep pie, make it the day you plan to serve it. You'll be less likely to get weeping due to meringue's natural tendency to break down.